Blow-up of solutions for wave equation with multiple ?(x)-Laplacian and variable exponent nonlinearities

Abstract

We consider an initial value problem related to the equation \begin{equation*} u_{tt}-{div}\left( \left\vert \nabla u\right\vert ^{m\left( x\right) -2}\nabla u\right) -{div}\left( \left\vert \nabla u_{t}\right\vert ^{r\left( x\right) -2}\nabla u_{t}\right) -\gamma \Delta u_{t}=\left\vert u\right\vert ^{p\left( x\right) -2}u, \end{equation*} with homogeneous Dirichlet boundary condition in a bounded domain $\Omega $. Under suitable conditions on variable-exponent $m\left( .\right) ,$ $r\left( .\right), $ and $p\left( .\right) ,$ we prove a blow-up of solutions with negative initial energy.